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An approximate universal coefficient theorem
2005
Transactions of the American Mathematical Society
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An approximate Universal Coefficient Theorem (AUCT) for certain C * -algebras is established. ...
We also show that C * -algebras that are locally approximated by C * -algebras satisfying the AUCT satisfy the AUCT. ...
Part of this work was done when the author was visiting East China Normal University. He thanks the Department of Mathematics for providing a shelter during the summer 2000 in Shanghai. ...
doi:10.1090/s0002-9947-05-03696-2
fatcat:jr5272drsnemtm7uqmw4tyvn7m
Universal Approximation Theorem for Neural Networks
[article]
2021
arXiv
pre-print
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Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks". ...
This paper is a comprehensive explanation of the universal approximation theorem for feedforward neural networks, its approximation rate problem (the relation between the number of intermediate units and ...
arXiv:2102.10993v1
fatcat:i3rbo7bs2jgvzlwhieapetitdi
Universal approximation theorem for Dirichlet series
2006
International Journal of Mathematics and Mathematical Sciences
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Several density results are proved that finally lead to the main theorem on simultaneous approximation. ...
The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right ...
The present paper can also be seen as an extension of [6] to universal Dirichlet series. Moreover, Theorem 4.3 is a universal approximation result in the sense of [7] . ...
doi:10.1155/ijmms/2006/37014
fatcat:737d3r4ktzdxrb4547e3jz6qt4
Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions
2013
Journal of Approximation Theory
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2012 pre-print arXiv:1010.0850v5 fatcat:bptik4dn2nguzportbagcg474mOther Versions
We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose ...
We apply this result on the Voronin universality theorem for compact sets K of this type, where the usual condition that the function is nonvanishing on the boundary can be removed. ...
By Conjecture 2 we can approximate f (z) by a polynomial p(z) such that
Relating Mergelyan's theorem and Voronin universality |p(z) − f (z)| < ε/2, (z ∈ K), (1) where p(z) is nonvanishing on K. ...
doi:10.1016/j.jat.2012.12.005
fatcat:zelqzp3lfnakbfooevr6lvs5pq
A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
[article]
2020
arXiv
pre-print
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This paper studies the universal approximation property of deep neural networks for representing probability distributions. ...
We prove upper bounds for the size (width and depth) of the deep neural network in terms of the dimension d and the approximation error ε with respect to the three discrepancies. ...
Our main result is the universal approximation theorem for expressing probability distributions. Theorem 2.1 (Main theorem). ...
arXiv:2004.08867v3
fatcat:bassh6lnu5czljl746ratj3eey
Universal Approximation Theorems for Differentiable Geometric Deep Learning
[article]
2022
arXiv
pre-print
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In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)'s approximation theorem and a data-dependent version of Yarotsky and Zhevnerchuk (2019)'s uncursed approximation ...
As applications, we confirm the universal approximation capabilities of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015 ...
These networks are obtained via the universal approximation theorem. Hence, to derive the estimates of Proposition 53, we should constructively prove the universal approximation theorem (Theorem 46). ...
arXiv:2101.05390v4
fatcat:dgbgiv7ymrbttapiwlgiizrtaa
The universal approximation theorem for complex-valued neural networks
[article]
2020
arXiv
pre-print
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We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. ...
We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function ...
Related work The classical universal approximation theorem There exist many versions of the universal approximation theorem for real networks. ...
arXiv:2012.03351v1
fatcat:3wq474t47vctre3oidrbpugo6a
Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks
[article]
2021
arXiv
pre-print
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The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. ...
Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators. ...
In the approximation theory of neural networks (NNs), universal approximation theorems (UATs) are statements that establish the density of a class of NNs within a space of mappings. ...
arXiv:2109.11354v1
fatcat:rtezxzkrubglzikwjikbgqoeji
A Universal Approximation Theorem for Mixture-of-Experts Models
2016
Neural Computation
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Our result can be viewed as a universal approximation theorem for MoE models. ...
The theorem we present allows MoE users to be confident in applying such models for estimation when data arise from nonlinear and nondifferentiable generative processes. ...
Our result is a universal approximation theorem, similar in spirit to Cybenko (1989, theorem 2) , where the linear combination of sigmoidal functions is proved dense in C(X). ...
doi:10.1162/neco_a_00892
pmid:27626962
fatcat:7gwgc5ahezgenmcftmjyl4dxca
Universal approximation theorem for uninorm-based fuzzy systems modeling
2003
Fuzzy sets and systems (Print)
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A Universal Approximation Theorem for Mixture of Experts Models
[article]
2016
arXiv
pre-print
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Our result can be viewed as a universal approximation theorem for MoE models. ...
Our result is a universal approximation theorem for MoE mean functions in the style of [3] . ...
Stone-Weierstrass Theorem Following the presentation of [2] , the Stone-Weierstrass Theorem can be phrased as follows. Theorem 1. ...
arXiv:1602.03683v1
fatcat:svv6rphry5fdjfj2rfqlujjjo4
Lavrentiev's approximation theorem with nonvanishing polynomials and universality of zeta-functions
[article]
2010
arXiv
pre-print
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We prove a variant of the Lavrentiev's approximation theorem that allows us to approximate a continuous function on a compact set K in C without interior points and with connected complement, with polynomial ...
We use this result to obtain a version of the Voronin universality theorem for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous ...
By Theorem 1 we can approximate f (z) by a polynomial g(z) such that |g(z) − f (z)| < ǫ/2, z ∈ K, (3) where g(z) is nonvanishing on K. ...
arXiv:1010.0386v1
fatcat:e3rwnxjmpnekphxsjajhflbdmi
Characterizing the Universal Approximation Property
[article]
2020
arXiv
pre-print
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Moreover, we show that most function spaces admit universal approximators built using a single function. ...
To better understand the approximation capabilities of various currently available neural network architectures, this paper studies the universal approximation property itself across a broad scope of function ...
Theorem 2.3 (Equivalence of Universal Approximators to the Feed-Forward Architecture). ...
arXiv:1910.03344v3
fatcat:j3mhfpj3mbf35iz2ypotvyt5de
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
[article]
2020
arXiv
pre-print
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This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. ...
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can ...
Acknowledgments We thank Yanhui Su of Fuzhou University for the help on Theorem 2. We thank Zhongqiang Zhang of Worcester Polytechnic Institute for the proof in Appendix C. ...
arXiv:1910.03193v3
fatcat:67kretzwczffriihix3zn7fs6i
A Gradient Free Neural Network Framework Based on Universal Approximation Theorem
[article]
2020
arXiv
pre-print
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We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding laborious iterations. ...
The method is based on the calculation of the weights of each neuron in small neighborhoods of the data, such that the corresponding local approximation matrix is invertible. ...
This complies with the Universal Approximation theorem, and offers a geometric point of view. ...
arXiv:1909.13563v3
fatcat:5u34jsdskvho7e5bc2fj3u6rai
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